Inductive reasoning is the process of making inferences based upon observed patterns, or simple repetition. Often used in reference to predictions about will happen or does happen, based upon what has happened.

The problem of induction is one of considerable debate and importance in the philosophy of science: is induction indeed justified, and if so, how?

History

Ancient philosophy

For a move from particular to universal, Aristotle in the 300s BCE used the Greek word epagogé, which Cicero translated into the Latin word inductio. In the 300s CE, Sextus Empiricus maintained that all knowledge derives from sensory experience—concluded in his Outlines of Pyrrhonism that acceptance of universal statements as true cannot be justified by induction.

Early modern philosophy

In 1620, early modern philosopherFrancis Bacon repudiated mere experience and enumerative induction, and sought to couple those with neutral and minute and many varied observations before to uncover the natural world's structure and causal relations beyond the present scope of experience via his method of inductivism, which nonetheless required enumerative induction as a component.

The supposedly radical empiricist David Hume's 1740 stance found enumerative induction to have no rational, let alone logical, basis but to be a custom of the mind and an everyday requirement to live, although observations could be coupled with the principle uniformity of nature—another logically invalid conclusion, thus the problem of induction—to seemingly justify enumerative induction and reason toward unobservables, including causality counterfactually, simply that modifying such an aspect prevents or produces such outcome.

Awakened from "dogmatic slumber" by a German translation of Hume's work, Kant sought to explain the possibility of metaphysics. In 1781, Kant's Critique of Pure Reason introduced the distinction rationalism, a path toward knowledge distinct from empiricism. Kant sorted statements into two types. The analytic are true by virtue of their terms' arrangement and meanings—thus are tautologies, merely logical truths, true by necessity—whereas the synthetic arrange meanings to refer to states of facts, contingencies. Finding it impossible to know objects as they truly are in themselves, however, Kant found the philosopher's task not peering behind the veil of appearance to view the noumena, but simply handling phenomena.

During the 1830s and 1840s, while Comte and Mill were the leading philosophers of science, William Whewell found enumerative induction not nearly so simple, but, amid the dominance of inductivism, described "superinduction". Whewell proposed recognition of "the peculiar import of the term Induction", as "there is some Conception superinduced upon the facts", that is, "the Invention of a new Conception in every inductive inference". Rarely spotted by Whewell's predecessors, such mental inventions rapidly evade notice. Whewell explained,

"Although we bind together facts by superinducing upon them a new Conception, this Conception, once introduced and applied, is looked upon as inseparably connected with the facts, and necessarily implied in them. Having once had the phenomena bound together in their minds in virtue of the Conception, men can no longer easily restore them back to detached and incoherent condition in which they were before they were thus combined".

These "superinduced" explanations may well be flawed, but their accuracy is suggested when they exhibit what Whewell termed consilience—that is, simultaneously predicting the inductive generalizations in multiple areas—a feat that, according to Whewell, can establish their truth. Perhaps to accommodate prevailing view of science as inductivist method, Whewell devoted several chapters to "methods of induction" and sometimes said "logic of induction"—and yet stressed it lacks rules and cannot be trained.

Contemporary philosophy

Bertrand Russell

Having highlighted Hume's problem of induction, John Maynard Keynes posed logical probability as its answer—but then figured not quite. Bertrand Russell found Keynes's Treatise on Probability the best examination of induction, and if read with Jean Nicod's Le Probleme logique de l'induction as well as R B Braithwaite's review of it in the October 1925 issue of Mind, to provide "most of what is known about induction", although the "subject is technical and difficult, involving a good deal of mathematics". Two decades later, Russell proposed enumerative induction as an "independent logical principle". Russell found,

"Hume's skepticism rests entirely upon his rejection of the principle of induction. The principle of induction, as applied to causation, says that, if A has been found very often accompanied or followed by B, then it is probable that on the next occasion on which A is observed, it will be accompanied or followed by B. If the principle is to be adequate, a sufficient number of instances must make the probability not far short of certainty. If this principle, or any other from which it can be deduced, is true, then the casual inferences which Hume rejects are valid, not indeed as giving certainty, but as giving a sufficient probability for practical purposes. If this principle is not true, every attempt to arrive at general scientific laws from particular observations is fallacious, and Hume's skepticism is inescapable for an empiricist. The principle itself cannot, of course, without circularity, be inferred from observed uniformities, since it is required to justify any such inference. It must therefore be, or be deduced from, an independent principle not based on experience. To this extent, Hume has proved that pure empiricism is not a sufficient basis for science. But if this one principle is admitted, everything else can proceed in accordance with the theory that all our knowledge is based on experience. It must be granted that this is a serious departure from pure empiricism, and that those who are not empiricists may ask why, if one departure is allowed, others are forbidden. These, however, are not questions directly raised by Hume's arguments. What these arguments prove—and I do not think the proof can be controverted—is that the induction is an independent logical principle, incapable of being inferred either from experience or from other logical principles, and that without this principle, science is impossible".